Hi Bo,
I have been married to a New Zealander for 25 years and we have done over 40 trips back to NZ. She has relatives we have visited all over NZ over the past 25 years. In addition we have done 4 major trips (>5 weeks) around NZ. Two in the North Island and two in the South. Notwithstanding my wife is from the North Island, we both prefer the South Island. The North Island is also very pretty with lots to see and do and you will have a great time. Driving in the North Island is definitely a bit easier than the South Island, as the highways are a little more open with fewer twisty, winding, hilly bits.
While 95% of the highest Mountains are in the South Island there are still probably 10 in the North Island over 5,000 feet. Ruapehu is the highest, some others are Taranaki (Egmont), Tongariro, Ngauruhoe, Hikurangi.
You have already been given some excellent destinations. It's also difficult to fit everything into the time you have.
Two other destinations I would include as "must see" are
1) Huka Falls (just north of Taupo)
https://www.newzealand.com/au/feature/huka-falls/
2) Wai O TaPu (About 30 minutes nth of Huka Falls and 40 minutes Sth of Rotorua)
https://www.waiotapu.co.nz/
If you also go to Napier / Hastings I would also do the drive up Te Mata Peak at Havelock Nth. Its a hairy drive up for the last couple of km's but the views at the top are spectacular
https://www.newzealand.com/au/feature/te-mata-peak/
You have indicated you are only going as far North as Auckland. Someone has suggested the Bay of Islands. This is about a 4 hour drive North of Auckland and to go up that far will take at least 2 full days for the round trip and sightseeing. I think The Bay of Islands is quite pretty and worth doing (I am a fisherman and the fishing is also excellent), but my wife who was born right near the Bay of Islands doesn't rate it quite so high.
Cheers and have a great trip
John B